Explicit Result on Equivalence of Rational Quadratic Forms Avoiding Primes

TL;DR

This paper presents an effective method to determine rational equivalence of quadratic forms over $Q$ that are locally equivalent at a specified finite set of primes, addressing a key question in the theory of quadratic forms.

Contribution

It provides a new, effective approach to decide rational equivalence of quadratic forms in the same genus, avoiding primes, which was previously unresolved.

Findings

An effective algorithm for rational equivalence over specified primes

Addresses a key open question in quadratic form theory

Connects local and global equivalence in quadratic forms

Abstract

Given a pair of regular quadratic forms over $\mathbb{Q}$ which are in the same genus and a finite set of primes $P$, we show that there is an effective way to determine a rational equivalence between these two quadratic forms which are integral over every prime in $P$. This answers one of the principal questions posed by Conway and Sloane in their book {\em Sphere packings, lattices and groups}, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol 290, Springer-Verlag, New York, 1999; page 402.